Name
MATH 308
ID
Exam I
Fall 2000
Section 200
P. Yasskin
1
/10 5
/10
2
/10 6
/15
3
/10 7
/20
4
/10 8
/15
HAND COMPUTATIONS
Problems 1-5:
Solve the initial value problem using the method appropriate to one of the
following types of first order differential equations:
a. Separable Equation
b. Equation with Homogeneous Coefficients
c. Linear Equation
d. Bernoulli Equation
Hint: k 1
1"s
Exact
Equation
e.
Be sure to identify the equation type. There is one problem for each equation type.
1. (10 points)
dy
4y x
dx
x
with
yŸ1 0
2. (10 points)
2y
dy
" t 6t y
dt
3. (10 points) dx x 12
dt
t
tx
with
with
yŸ1 9
xŸ1 2
4. (10 points)
dy
sin x " sin y
xycos
y cos x
dx
5. (10 points)
2
dy
x2 y
dx
y
x
with
with
yŸ0 2
yŸ1 3
6. (15 points) Set up the differential equation and initial condition for PŸt in the following
problem. Do not solve the equations.
A certain pond can contain 12, 000 ft 3 of water before the dam overflows. Initially, there
are 8, 000 ft 3 of water and 50 gallons of pollution in the pond. Acme Polluters is putting
2 gallons of pollution in the pond a day. Every day, 2, 000 ft 3 of fresh water is pumped
into the pond and 1, 000 ft 3 of polluted water is pumped out. Let PŸt be the gallons of
pollution in the pond after t days. When does the dam overflow and what is the
concentration of the polution in the water when the dam first overflows?
MAPLE COMPUTATIONS
dy
e t " y.
7. (20 points) Consider the differential equation
dt
a. (4 pts) Find the general solution.
b. (4 pts) Find the specific solution satisfying the initial condition yŸ0 4.
c. (4 pts) Plot the direction field of the differential equation for times "4 t t t 6.
d. (4 pts) Add to the direction field the solutions satisfying each of the initial
conditions
yŸ0 "4, yŸ0 "2, yŸ0 0, yŸ0 2, yŸ0 4, yŸ0 6
Adjust the vertical range to a reasonable value.
e. (4 pts) Looking at the exact solution and the plot, describe the behavior of the
solutions for large times. At about what time does this asymptotic behavior begin?
dy
cos x sin y with yŸ0 1.
dx
a. (10 pts) What happens if you try to solve the equations using dsolve ? Find a
Taylor polynomial approximation about x 0 for the solution to this initial value
problem keeping terms up to and including x 4 .
b. (5 pts) Plot the direction field for this differential equation together with the solution
satisfying the initial condition. Plot the Taylor polynomial approximation. Combine
the two plots into a single plot. On approximately what interval is the Taylor
polynomial a good approximation?
8. (15 points) Consider the initial value problem
To Turn in Your Maple Computations:
1. Save your Maple file as lastname_exam1.mws
2. Print your file as follows:
a. Click on FILE, PRINT and Printer Command.
lpr -J ” Yasskin Maple Exam 1”
b. Make the command read:
c. Call Dr. Yasskin over to check your printing.
d. Click on PRINT.
3. Mail your file as follows:
pine
a. Start the mail program:
C.
b. Compose a letter by typing
c. In the header region, enter:
To
yasskin
Attachment
lastname_exam1.mws (or the exact name of your Maple file)
Subject
Last Name
Exam1
d. Call Dr. Yasskin over to check your email.
^X
and
Y.
e. Mail the letter by typing